Variance and Standard Deviation

1Summary 重點摘要

Learn how to measure spread of data — variance and standard deviation.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: The meaning of dispersionThe mean describes only the centre of a data set, not how spread out the values are; measures of spread are called dispersion, and variance and standard deviation are the two most common. For example, two classes may both average 70, yet one has scores packed between 65 and 75 while the other ranges from 30 to 100, very different in consistency. Why is dispersion needed? With the mean alone the two classes look identical. A common mistake is comparing data sets by the mean only, ignoring the spread entirely.
2. Step 2: The definition of varianceVariance is obtained by adding the squared differences between each value and the mean, then dividing by the number of values: Var = Σ(x − mean)² ÷ n. Why square? Deviations are positive and negative and would cancel if added directly, for example −2 and 2 sum to 0 and hide the spread; squaring makes all terms positive so the sizes accumulate. Why divide by n at the end? It averages the squared total, allowing fair comparison between groups. A common mistake is summing without squaring, which always gives 0 or near 0; a zero result signals the error at once.
3. Step 3: Computing the deviationsTake the data 2, 4, 6 as a full demonstration: first find the mean, (2 + 4 + 6) ÷ 3 = 4; then find each deviation, each value minus the mean: 2 − 4 = −2, 4 − 4 = 0, 6 − 4 = 2. Why is the mean found first? Variance measures how far values stray from the mean, so the mean is the reference point of the whole calculation. A common mistake is subtracting the largest value or the first value instead of the mean; once the reference is wrong everything after fails. As a check, the deviations should sum to 0.
4. Step 4: Squaring and finding the varianceSquare each deviation from the previous step: (−2)² = 4, 0² = 0, 2² = 4; add them to get 4 + 0 + 4 = 8; finally divide by the number of values 3, giving variance = 8 ÷ 3 ≈ 2.67. Why does a negative deviation become positive? Squaring removes the sign, so (−2)² and 2² are both 4, and direction does not affect size. A common mistake is mishandling the negative sign, writing (−2)² = −4; another is dividing by n − 1, which belongs to another statistical setting, while school courses divide by n.
5. Step 5: From variance to standard deviationThe standard deviation is the square root of the variance: standard deviation = √variance. In the example the variance is 83 ≈ 2.67, so the standard deviation = √(83) ≈ 1.63. Why take the root? Squaring changes the units, so if the data are in centimetres the variance is in square centimetres; the square root returns the unit to centimetres, allowing direct comparison with the original data. A smaller standard deviation means the data cluster near the mean. A common mistake is confusing variance with standard deviation; the two differ, and the question must be read for which one is asked.
6. Step 6: Practical useWhen comparing two data sets, the standard deviation is the key to judging consistency: a small standard deviation in group A means values hug the mean and performance is steady; a large standard deviation in group B means values swing widely. For example, two students both average 80 over several tests, but with standard deviations of 3 and 12; the first is consistent while the second fluctuates. Why compare standard deviations when the means are equal? Consistency is information the mean cannot show. A common mistake is judging by a single best score; examine both the mean and the standard deviation.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Variance = average of squared deviations from mean
• Standard deviation = √variance
• Smaller SD → more clustered, more consistent
• Larger SD → more spread, more inconsistent
• Used to compare consistency of two datasets